Quantum automata and quantum grammars
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Quantum Automata and Quantum Grammars
Cristopher Moore and Jamta Fe Institute, 1399 Hyde Park Road, Santa Fe NM 87501 USA
fmoore,jpcg@
1 Introduction
Nontraditional models of computation | such as real-valued, analog, spatial, molecular, stochastic, and quantum computation | have received a great deal of interest in both physics and computer science in recent years (e.g. 1, 4, 10, 21, 8, 31, 9]). This stems partly from a desire to understand computation in dynamical systems, such as ordinary di erential equations, iterated maps, cellular automata, and recurrent neural networks, and partly from a desire to circumvent the fundamental limits on current computingx technologies by inventing new computational model classes. Quantum computation, in particular, has become a highly active research area. This is driven by the recent discovery of quantum algorithms for factoring that operate in polynomial time 29], the suggestion that quantum computers can be built using familiar physical systems 7, 14, 19], and the hope that errors and decoherence of the quantum state can be suppressed so that such computers can operate for long times 30, 33]. If we are to understand computation in a quantum context, it might be useful to translate as many concepts as possible from classical computation theory into the quantum case. From a practical viewpoint, we might as well start with the lowest levels in the computational hierarchy and work upward. In this paper we begin in just this way by de ning quantum versions of the simplest language classes | the regular and context-free languages 16]. To do this, we de ne quantum nite-state and push-down automata (QFAs and QPDAs) as special cases of a more general object, a real-time quantum automaton. In this setting a formal language becomes a function that assigns quantum probabilities to words. We also de ne quantum grammars, in which we sum over all derivations to nd the amplitude of a word. We show that the corresponding languages, generated by quantum grammars and recognized by quantum automata, have pleasing properties in analogy to their classical counterparts. These properties include pumping lemmas, closure properties, rational and (almost) algebraic generating functions, and Greibach normal form. For the most part, our proofs simply consist of tracking standard results in the theory of classical languages and automata, stochastic automata, and formal power series, and attaching complex amplitudes to the transitions and productions of our automata and grammars. In a few places | notably, lemmas 12 and 13 and theorems 6, 7, 10, 19, 23, and 24 | we introduce genuinely new ideas. We believe that this strategy of starting at the lowest rungs of the Chomsky hierarchy has several bene ts. First, we can make concrete comparisons between classical and quantum computational models. This comparison is di cult to make for more powerful models, because of unsolved problems in computer science (for instance, deterministic vs. quantum polynomial time). Second, studying the computational power of a physical system can give detailed insights into a natural system's structure and dynamics. For example, it may be the case that the spatial density of physical computation is nite. In this case, every nite quantum computer is actually a QFA. If a system does in fact have in nite memory, it makes sense to ask what kinds of long-time correlations it can have, such as whether its memory is stack-like or queue-like. Our QPDAs provide a way to formalize these questions.
2
Cris Moore and James P. Crutch eld
Molecular biology suggests another example along these lines, the class of protein secondary structures coded for by RNA. To some approximation the long-range correlations between RNA nucleotide base pairs responsible for secondary structure can be modeled by parenthesis-matching grammars 28, 27]. Since RNA macromolecules are quantum mechanical objects, constructed by processes that respect atomic and molecular quantum physics, the class of secondary structures coded for by RNA may be more appropriately modeled by the quantum analogs of context-free grammars introduced here. In the same vein, DNA and RNA nucleotide sequences are recognized and manipulated by various active molecules (e.g. transcription factors and polymerases), could their functioning be modeled by QFAs and QPDAs? Finally, the theory of context-free languages has been extremely useful in designing compilers, parsing algorithms, and programming languages for classical computers. Is it possible that quantum context-free languages can play a similar role in the design of quantum computers and algorithms?
Abstract. To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case. To that end, we propose quantum versions of nite-state and push-down automata, and regular and context-free grammars. We nd analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form. We also show that there are quantum context-free languages that are not context-free.
Cristopher Moore and Jamta Fe Institute, 1399 Hyde Park Road, Santa Fe NM 87501 USA
fmoore,jpcg@
1 Introduction
Nontraditional models of computation | such as real-valued, analog, spatial, molecular, stochastic, and quantum computation | have received a great deal of interest in both physics and computer science in recent years (e.g. 1, 4, 10, 21, 8, 31, 9]). This stems partly from a desire to understand computation in dynamical systems, such as ordinary di erential equations, iterated maps, cellular automata, and recurrent neural networks, and partly from a desire to circumvent the fundamental limits on current computingx technologies by inventing new computational model classes. Quantum computation, in particular, has become a highly active research area. This is driven by the recent discovery of quantum algorithms for factoring that operate in polynomial time 29], the suggestion that quantum computers can be built using familiar physical systems 7, 14, 19], and the hope that errors and decoherence of the quantum state can be suppressed so that such computers can operate for long times 30, 33]. If we are to understand computation in a quantum context, it might be useful to translate as many concepts as possible from classical computation theory into the quantum case. From a practical viewpoint, we might as well start with the lowest levels in the computational hierarchy and work upward. In this paper we begin in just this way by de ning quantum versions of the simplest language classes | the regular and context-free languages 16]. To do this, we de ne quantum nite-state and push-down automata (QFAs and QPDAs) as special cases of a more general object, a real-time quantum automaton. In this setting a formal language becomes a function that assigns quantum probabilities to words. We also de ne quantum grammars, in which we sum over all derivations to nd the amplitude of a word. We show that the corresponding languages, generated by quantum grammars and recognized by quantum automata, have pleasing properties in analogy to their classical counterparts. These properties include pumping lemmas, closure properties, rational and (almost) algebraic generating functions, and Greibach normal form. For the most part, our proofs simply consist of tracking standard results in the theory of classical languages and automata, stochastic automata, and formal power series, and attaching complex amplitudes to the transitions and productions of our automata and grammars. In a few places | notably, lemmas 12 and 13 and theorems 6, 7, 10, 19, 23, and 24 | we introduce genuinely new ideas. We believe that this strategy of starting at the lowest rungs of the Chomsky hierarchy has several bene ts. First, we can make concrete comparisons between classical and quantum computational models. This comparison is di cult to make for more powerful models, because of unsolved problems in computer science (for instance, deterministic vs. quantum polynomial time). Second, studying the computational power of a physical system can give detailed insights into a natural system's structure and dynamics. For example, it may be the case that the spatial density of physical computation is nite. In this case, every nite quantum computer is actually a QFA. If a system does in fact have in nite memory, it makes sense to ask what kinds of long-time correlations it can have, such as whether its memory is stack-like or queue-like. Our QPDAs provide a way to formalize these questions.
2
Cris Moore and James P. Crutch eld
Molecular biology suggests another example along these lines, the class of protein secondary structures coded for by RNA. To some approximation the long-range correlations between RNA nucleotide base pairs responsible for secondary structure can be modeled by parenthesis-matching grammars 28, 27]. Since RNA macromolecules are quantum mechanical objects, constructed by processes that respect atomic and molecular quantum physics, the class of secondary structures coded for by RNA may be more appropriately modeled by the quantum analogs of context-free grammars introduced here. In the same vein, DNA and RNA nucleotide sequences are recognized and manipulated by various active molecules (e.g. transcription factors and polymerases), could their functioning be modeled by QFAs and QPDAs? Finally, the theory of context-free languages has been extremely useful in designing compilers, parsing algorithms, and programming languages for classical computers. Is it possible that quantum context-free languages can play a similar role in the design of quantum computers and algorithms?
Abstract. To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case. To that end, we propose quantum versions of nite-state and push-down automata, and regular and context-free grammars. We nd analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form. We also show that there are quantum context-free languages that are not context-free.